# Structure closed under unary operation and isomorphism thereof.

There are several answers regarding this on the site but none of them seem to be in consensus. Some call it a function, other unoid and some people something else. So I decided to ask it with a example.

Consider the set

$$S_1 = \{ 2, 5, 42, 10, 45, 4, 12, 60, 93\}$$

and the unary operation

$$f_1 : S_1 \to S_1, x \mapsto 26(x^2 +8x)\mod 103$$

in other words,

$$f_1 (x) = 26(x^2+8x) \mod 103$$

As we can see $$S_1$$ is closed under unary operation $$f_1$$. So like groups (set closed under binary operation satisfying certain axioms) is there a name for this structure?

Addittionally, let us consider the set

$$S_2 = \{2, 4, 8, 16, 32, 64, 55, 37, 1\}$$

and the following unary operation:

$$f_2 : S_2 \to S_2, x \mapsto 2x\mod 73$$,

in other words,

$$f_2 (x) = 2x \mod 73$$

Since cardinality of both structures are $$9$$ is it possible that they are isomorphic to each other. If so then how could such isomorphism be found?

• You might be interested in universal algebra. Commented Jan 27 at 12:54
• For every set $S$ you can define a (surjective) map $f\colon S\to S$. But with this only, $S$ is left to be just a... set. And we don't say by this that "$S$ is closed under $f$", to my knowledge. Commented Jan 27 at 13:01
• Let $f_1:X\to X$ and $f_2:Y\to Y$ be bijections and $|X|=|Y|$. The unary algebras $(X,f_1)$ and $(Y,f_2)$ are isomorphic if and only if $f_1$ and $f_2$ have the same cyclic type. Commented Jan 27 at 14:46

This corresponds to a directed graph with outdegree $$1$$. (I don't assume the operation to be surjective and/or bijective because the question doesn't say so.) Such graphs are called functional graphs because, yes, this is just a set and a function from that set to that set. Accordingly, you can apply tools for detecting graph isomorphisms to this problem.